Finite Group Databasebeta

← #27362615

Subgroups — #27362615

Order 13568.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3#20284187C211yesmaximal, characteristic, complemented
4#20273929C214yesmaximal, complemented
5#20273787C214yesmaximal, complemented
6#20273396C211yesmaximal, characteristic, complemented
7#20273140C211yesmaximal, characteristic, complemented
8#20273110C211yesmaximal, characteristic, complemented
9C2×C53×#554C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16810997C2212yescomplemented
13#16810973C2212yescomplemented
14#16810936—21no—
15#16810903C2212yes—
16#16810845—22no—
17#16810833C2212yescomplemented
18#16810825—22no—
19#16810823C2212yescomplemented
20#16810818C2212yes—
21#16810762—22no—
22#16810757—21no—
23C4×#11719922—22no—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810622—21no—
28#16810372C2214yescomplemented
29Q8⋊#88322C2214yescomplemented
30D4⋊#88322C2214yes—
31C53×#554—22no—
32C53×#554—22no—
33C53×#554—22no—
34C53×#554C2214yes—
35C2×C4⋊Q8×C53C2211yescharacteristic
36C2×#125×C53—21no—
37C2×C22⋊Q8×C53—21no—
38C2×C22⋊Q8×C53—22no—
39C2×C4×Q8×C53C2211yescharacteristic
40C2×C4×D4×C53—21no—
41C2×C4×D4×C53C2211yescharacteristic
42C22×C4⋊C4×C53—21no—
43C2×#11719945—21no—
44#13133248—44no—
45C2×#11719925—22no—
46C2×#11719924—22no—
47C2×#11719924—24no—
48C2×#11719922—22no—
49C2×#11719917—21no—
50#13133217C2311yescharacteristic
51#13133214C2311yescharacteristic
52#13133205—21no—
53#13133191—21no—
54#13133191—21no—
55C4×D212—21no—
56C4×D212—21no—
57#13133150—22no—
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—24no—
63#13133140—22no—
64#13133138—22no—
65#13133137—22no—
66#13133137—22no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69C4×#88322—21no—
70C4×#88322C2312yes—
71C22×Q8×C53D411yescharacteristic
72C22×Q8×C53D412yes—
73C22×D4×C53D411yescharacteristic
74C23×C4×C53—22no—
75C23×C4×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53D412yes—
79C4⋊Q8×C53C2312yes—
80#125×C53—24no—
81#125×C53—24no—
82C22⋊Q8×C53—28no—
83C22⋊Q8×C53—24no—
84C22⋊Q8×C53—28no—
85C22⋊Q8×C53—24no—
86C4×Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53D412yes—
89C4×Q8×C53C2312yes—
90C4×D4×C53—24no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—24no—
94C4×D4×C53D412yes—
95C4×D4×C53C2312yes—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—24no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C24×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—82no—
115C2×Q8×C53C2×D412yes—
116C2×Q8×C53C2×D414yes—
117C2×D4×C53—24no—
118C2×D4×C53—21no—
119C2×D4×C53C2×D412yes—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—28no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—84no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C2×C22⋊D53—41no—
134#11719945—42no—
135#11719945—42no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×C4×D53—41no—
140C2×Dic106—21no—
141C4⋊C4×C53—28no—
142C4⋊C4×C53—28no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53C2×D412yes—
152C4⋊C4×C53C2×D412yes—
153C22⋊C4×C53—28no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C42×C53—22no—
157C42×C53—22no—
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53C2×D411yescharacteristic
161C42×C53C2411yescharacteristic
162#11719925—44no—
163#11719924—44no—
164#11719924—84no—
165#11719922—44no—
166#11719921—41no—
167#11719921—41no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170#11719917—42no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C23×C53—24no—
175C23×C53—22no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53#12011yescharacteristic
179C22×D53—41no—
180Q8×C53—22no—
181Q8×C53—22no—
182Q8×C53—44no—
183Q8×C53—42no—
184Q8×C53—82no—
185D4×C53—24no—
186D4×C53—22no—
187D4×C53#13714yescomplemented
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—41no—
194C2×C4×C53—42no—
195C2×C4×C53—44no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—44no—
201C2×C4×C53—88no—
202C2×C4×C53#12011yescharacteristic
203C2×C4×C53#12011yescharacteristic
204C2×C4×C53#12011yescharacteristic
205C2×C4×C53#13512yes—
206C2×C4×C53AGL(1,ℤ/8)12yes—
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C22⋊D53—82no—
211C2×Dic53—41no—
212D212—42no—
213D212—41no—
214D212—41no—
215D212—42no—
216C4×D53—82no—
217Dic106—41no—
218Dic106—41no—
219#88322—42no—
220#88322—42no—
221#88322—42no—
222#18752—531nomaximal, Sylow
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—24no—
226C22×C53—48no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53#44912yes—
230C22×C53C2×#12011yescharacteristic
231C2×D53—81no—
232C2×D53—81no—
233C2×D53—81no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—42no—
238C4×C53—42no—
239C4×C53—42no—
240C4×C53—44no—
241C4×C53#44912yes—
242C4×C53C2×#13511yescharacteristic
243C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
244C4×C53C2×#13712yes—
245Dic53—81no—
246C2×#554D5311yescharacteristic, complemented
247#3206—531no—
248#3100—531no—
249#3037—531no—
250#1644—534no—
251#1622—534no—
252#1548—531no—
253#1486—531no—
254#1479—531no—
255C2×C53—22no—
256C2×C53—44no—
257C2×C53C2×#44911yescharacteristic
258C2×C53#300411yescharacteristic
259C2×C53#301911yescharacteristic
260D53—161no—
261C2×#137—1061no—
262#554—22no—
263#554—22no—
264#554—22no—
265#554C2×D5314yes—
266#532—531no—
267C2×C4⋊Q8C2×D5311yescharacteristic
268C2×#125—21no—
269C2×C22⋊Q8—21no—
270C2×C22⋊Q8—22no—
271C2×C4×Q8C2×D5311yescharacteristic
272C2×C4×D4—21no—
273C2×C4×D4C2×D5311yescharacteristic
274C22×C4⋊C4—21no—
275#492—532no—
276#487—532no—
277#480—1061no—
278#474—532no—
279#464—1062no—
280#462—532no—
281#461—1062no—
282#460—532no—
283#459—532no—
284#448—1062no—
285#447—1061no—
286C4×D8—1062no—
287#431—531no—
288#431—531no—
289#423—531no—
290#417—1061no—
291#333—534no—
292Q8⋊C8—534no—
293D4⋊C8—534no—
294C53#1875211yescharacteristic, complemented, Sylow
295C2×#41—1061no—
296C22×Q8C22⋊D5311yescharacteristic
297C22×Q8C22⋊D5312yes—
298C22×D4C22⋊D5311yescharacteristic
299C23×C4—22no—
300C23×C4—21no—
301#137—2124no—
302C2×Q16—1062no—
303C2×SD16—1064no—
304C2×SD16—1062no—
305C2×D8—1062no—
306C2×M16—1061no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—531no—
310C4⋊Q8C22⋊D5312yes—
311C4⋊Q8C22×D5312yes—
312#127—531no—
313#125—24no—
314#125—24no—
315#124—1061no—
316C22⋊Q8—24no—
317C22⋊Q8—28no—
318C22⋊Q8—24no—
319C22⋊Q8—28no—
320#121—1061no—
321#121—1061no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8C22⋊D5312yes—
325C4×Q8C22×D5312yes—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1061no—
331C4×D4—1061no—
332C4×D4C22⋊D5312yes—
333C4×D4C22×D5312yes—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4C22⋊D5311yescharacteristic
342C2×C4⋊C4C22⋊D5311yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C42—21no—
346C2×C42C22×D5311yescharacteristic
347#107—1062no—
348C4⋊C8—532no—
349C4⋊C8—532no—
350C4⋊C8—1061no—
351C4⋊C8—1061no—
352#103—1064no—
353#103—1062no—
354#102—1062no—
355#102—1062no—
356#102—1062no—
357C22⋊C8—1061no—
358C22⋊C8—1061no—
359C4×C8—532no—
360C4×C8—1061no—
361C24—21no—
362#41—2122no—
363#41—2122no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—82no—
369C2×Q8—1061no—
370C2×Q8C2×C22⋊D5312yes—
371C2×Q8C2×C22⋊D5314yes—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—1061no—
375C2×D4—1061no—
376C2×D4—1061no—
377C2×D4—2121no—
378C2×D4C2×C22⋊D5312yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—84no—
389C22×C4—2121no—
390C22×C4C2×C22⋊D5311yescharacteristic
391C22×C4C2×C22⋊D5311yescharacteristic
392C22×C4C2×C22⋊D5311yescharacteristic
393Q16—2124no—
394SD16—4244no—
395SD16—2124no—
396D8—2124no—
397M16—2122no—
398C2×C8—1062no—
399C2×C8—1062no—
400C2×C8—1062no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4C2×C22⋊D5312yes—
414C4⋊C4C2×C22⋊D5312yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C42—22no—
421C42—22no—
422C42C2×C22⋊D5311yescharacteristic
423C42C2×C22⋊D5311yescharacteristic
424C42C2×C22⋊D5311yescharacteristic
425C42C23×D5311yescharacteristic
426C23—24no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—2121no—
431C23#1313318611yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—44no—
435Q8—42no—
436Q8—82no—
437Q8—2121no—
438Q8—2121no—
439D4—24no—
440D4—22no—
441D4—4242no—
442D4—2122no—
443D4—2121no—
444D4—2121no—
445D4—2122no—
446D4#1313324814yescomplemented
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—88no—
461C2×C4—4242no—
462C2×C4—2121no—
463C2×C4#1313318611yescharacteristic
464C2×C4#1313318611yescharacteristic
465C2×C4#1313318611yescharacteristic
466C2×C4#1313323712yes—
467C2×C4#1313324212yes—
468C2×C4C22×C22⋊D5311yescharacteristic
469C2×C4C22×C22⋊D5311yescharacteristic
470C2×C4C22×C22⋊D5311yescharacteristic
471C8—2122no—
472C8—2122no—
473C8—2122no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—4241no—
481C22—4241no—
482C22—4241no—
483C22#1681076812yes—
484C22C2×#1313318611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—4241no—
493C4#1681076812yes—
494C4C2×#1313323711yescharacteristic
495C4C2×#1313324211yescharacteristic
496C4C2×#1313324812yes—
497C2—22no—
498C2—44no—
499C2—8481no—
500C2C2×#1681076811yescharacteristic
501C2#2028394911yescharacteristic
502C2#2028407211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#554C2
socleC22×C53C2×#120
prime radical#20284715C2
hypercenterC2×#554D53
hypocenterC53#18752
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20285640C2semidirect
3. #20284187C2semidirect
4. #20273929C2semidirect
5. #20273787C2semidirect
6. #20273396C2semidirect
7. #20273140C2semidirect
8. #20273110C2semidirect
9. C2×C53×#554C2semidirect
12. #16810997C22semidirect
13. #16810973C22semidirect
17. #16810833C22semidirect
19. #16810823C22semidirect
28. #16810372C22semidirect
29. Q8⋊#88322C22semidirect
187. D4×C53#137semidirect
246. C2×#554D53semidirect
294. C53#18752semidirect
446. D4#13133248semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202856402
#202847152
#202841872
#202739292
#202737872
#202733962
#202731402
#202731102
C2×C53×#5542
#1875253